SPIE 2010 - SALSA - 3D reconstruction - Bossa Nova Technologies

3D shape reconstruction of optical element using polarization
M. Vedel a , N. Lechocinski a , S. Breugnot a
a Bossa Nova Technologies, 606 Venice Blvd, Suite B, Venice, CA90291, USA
ABSTRACT
We present a novel polarization based metrological method of 3D shape measurement for in-line control of
optical surfaces and control of highly aspheric optical surfaces. This technique is fast, non contact, high resolution,
alignment free and with unprecedented dynamic. It has the potential to reach tens of nanometers accuracy. In this paper
we show that a polarization imaging camera combined with an un-polarized illumination and 3D reconstruction
algorithm lead to the 3D reconstruction of optical element (regular lens and aspheric lens) and the measurement of their
parameters. The optical element to be measured is placed in a diffusive integrating sphere and illuminated by unpolarized
light. The reflection of the un-polarized light by the optical element gets partially polarized. A polarization
camera captures the image of the optical element and measures the polarization state of each pixel in real time. The
Degree Of Light Polarized and the Angle Of Polarization parameters are related to the geometry of the optical element.
The 3D shape of the optical element is reconstructed using dedicated software. The architecture of the hardware,
calibration results and sensitivity measurements is presented and experimental results and observations as well as
possible further steps and new applications are discussed.
Keywords: 3D reconstruction, polarization imaging, linear Stokes parameters, metrology
1. INTRODUCTION
In this paper, we describe the results of a feasibility study for the development of a polarization based metrological
method of 3D shape measurement for in-line control of optical surfaces and control of highly aspheric optical surfaces.
The method uses the “Shape from Polarization” method developed by Miyazaki [1-3], Rahman [4-6] and Atkinson [7-9]
among others for the reconstruction of dielectric surfaces. This technique is based on the polarization imaging of the
optical element, leading to the measurement of the polarization state for each pixel of the image. This allows the
reconstruction of the 3D shape of the imaged optical element.
With increasing requirement for improving productivity and competitiveness, manufacturing companies are looking to
implement more in-line inspection techniques where the immediate feedback can lead to tighter tolerance, reduction in
scraps and reduction in energy consumption. In this paper, we show that a fast polarization imaging camera combined
with an un-polarized illumination and a 3D reconstruction algorithm led to the 3D reconstruction of optical element
(regular lens and aspheric lens) and the measurement of their parameters.
The current measurement methods (interferometers, contact profilometers) of optical surfaces are not adapted to high
volume as they are either fast but require precise alignment (optical methods) or not sensitive to alignment but very slow
(mechanical methods). Tomorrow’s optical surfaces will be very aspheric and have high numerical aperture to meet the
requirements of smaller and more cost effective optics. The current optical production is currently shifting toward these
lenses which are difficult or too slow to control with current methods. Using a fast polarization camera [16] instead of a
rotating mechanical polarizer allows performing the 3D reconstruction in less than 30 seconds.
In the developed polarization method of 3D shape measurement, the optical element to be measured is placed in a
diffusive integrating sphere and illuminated by un-polarized light. The reflection of the un-polarized light by the optical
element gets partially polarized according to Fresnel’s laws [12,13]. A polarization camera captures the image of the
optical element and measures the polarization state of each pixel in real time. The polarization state being related to the
geometry of the optical element, the 3D shape of the optical element can be reconstructed using dedicated software.
This paper starts with a theoretical paragraph to remind the principle of the method and the main tasks to be carried out.
Then the developed setup, software and the several tests performed to evaluate their performances are described. Finally
we discuss these results and the further steps to be addressed.

2. 3D RECONSTRUCTION BASED ON POLARIZATION IMAGING
2.1 Description of polarization
Light can be totally polarized, unpolarized or partially polarized. A totally polarized light can then be linearly
polarized, circularly polarized, or elliptically polarized. An unpolarized light is a light which has no coherence between
the electric fields along to orthogonal directions. In between light can be partially polarized. The polarization state of
light is usually described using Stokes formalism (Equation 1) [13].
⎡S
S
S r
=
S
⎢⎢⎢⎢
⎣S
0
1
2
3
⎤
⎥⎥⎥⎥
⎦
S
S
S
S
0
1
2
3
=
=
=
=
E E
x
E E
x
E E
x
x
*
x
*
x
*
y
j E E
+ E E
*
y
y
− E E
y
+ E E
y
y
*
y
*
y
*
x
− E E
*
x
= I
= I
= I
o
0
o
0
o
45
= I
+ I
− I
lh
o
90
o
90
− I
− I
o
−45
rh
(1)
It can describe any type of polarization and is linear with the intensity of light which makes it compatible with linear
algebra. The Stokes-Mueller formalism describes mathematically the polarization of light as well as its evolution. This
formalism uses real quadratic values (intensities along horizontal, vertical axes...) directly measured by detectors. A
Stokes vector is a 4 components vector. However, polarization is generally described with more physical parameters.
The parameters relevant for this paper are the Degree Of Linear Polarization (DOLP) and Angle Of linear Polarization
(AOP) (Equation 2). DOLP represents the relative amount of light linearly polarized; AOP represents the orientation of
this linearly polarized part.
DOLP
=
S
1
+ iS
S
0
2
2.2 Fresnel equations
1
AOP = Arg( S1
+ iS
2
)
(2)
2
Generally speaking, polarization of light will be modified after interacting with an object [12,13]. The possible
modifications include, but are not limited to, depolarization of a polarized light and polarization of an unpolarized light.
Polarization of an unpolarized light is a very general phenomenon. It comes from a difference in the reflection
coefficient for the electric field in the plane of incidence and perpendicular to the plane of incidence. This difference in
the reflection coefficient is perfectly determined by the angle of incidence on the surface and the physical properties of
the material. The reflection coefficient for the electric field in the plane of incidence (p-polarization) and perpendicular
to the plane of incidence (s-polarization) are calculated using the Fresnel equations [12]. The fact that the reflection
coefficients are not the same makes that an unpolarized light will become partially s-polarized after a reflection by a
dielectric surface (figure 1). Using the Fresnel equations, the DOLP can be calculated for any angle of incidence θ
(Equation 3). The calculation of the DOLP after a specular reflection of light by a dielectric surface shows that a
measurement of DOLP can be inverted to two possible angles of incidence with the hypothesis of a known material
refractive index and unpolarized incident light [1-11]. The same calculation can be conducted for other surfaces
materials. Morel described an extension of the “Shape from Polarization” method to metallic surfaces using a complex
refractive index [10,11].

2 2 2 2 2 4
2sin θ n −sin
θ −n
sin θ −sin
θ
DOLP reflection
=
(3)
2 2 2 2
4
n −sin
θ − n sin θ − 2sin θ
Figure 1. Fresnel laws: light reflected by a dielectric surface becomes partially polarized depending on the incidence angle
2.3 Shape from polarization
Gradient field estimation
Given a conventional Cartesian coordinate system (X,Y,Z) one can express the 3D shape by the height Z in
function of (X,Y) parameters. The method presented in this paper is based on the surface gradient field reconstruction
and its numerical integration. Figure 2 shows the geometrical parameters. Each normal vector (Equation 3) is expressed
knowing the local slope, related to the zenith angle θ i (incidence angle of the light), and the azimuth angle φ [1-11].
Partially
polarized light
DOLP(θ i )
AOP(φ)
X
N r Unpolarized
θ r illumination
θ i
φ
Y
⎛ ∂Z(
X , Y ) ⎞
−
⎛ p = tan( θ )cos( ϕ)
⎞ ∂
r ∂Z(
X , Y )
= = tan( )sin( ) = −
⎜⎜⎜
⎟⎟⎟
∂Y
⎝ 1 ⎠ 1
⎜⎜⎜⎜⎜⎜
X (3)
N q θ ϕ
⎟⎟⎟⎟⎟⎟
⎝
⎠
Figure 2. Geometrical parameters

Considering a specular reflection on the surface element the incidence angle θ i and the reflected angle θ r are the same.
We will denote it the zenith angle θ. The Fresnel equations link the surface’s local slope (zenith angle), the refractive
index n of the material and the Degree Of Linear Polarization (Equation 3). This equation can be inverted to express the
two zenith angles θ possible: one below and one above the Brewster incidence angle (figure 1). In most of the cases, the
first root will be the right one. For a usual glass with refractive index around 1.5, Brewster’s angle is 56°. Most of the
optics do not have such high slope. Even high numerical apertures aspherics or condensers have slopes below 45° at their
edge. The results presented in this paper are limited to these cases.
The second parameter φ is related to the Angle Of Polarization. As the polarization vector is defined within [0°; 180°]
cycle there are two possible azimuth angles for the normal vector (Equation 4).
ϕ = AOP or ϕ = AOP + π
(4)
Several methods to resolve that ambiguity have been described in previous papers [1-11]. A simple algorithm has been
used to disambiguate the normal to the surface. As shown in figure 3, for simple lenses shape, the normal to the surface
appear to radiate for the point of the lens oriented toward the lens. The angle θ also increases with the radius. These are
the properties used to disambiguate the normal to the surface for simple shape lenses: convex or concave surfaces with
no inflexion point.
Gradient field integration
Once the zenith and the azimuth angles are known for each pixel of the imaged sample we need to operate a
numerical integration in order to obtain the 3D height map Z(X,Y). Many off the shelf algorithms to project a gradient
field onto the closest integrable gradient field and to recover the 3D surface associated with this gradient field exist. The
most used is the Frankot-Chellapa algorithm [14]. The non integrable gradient field measured is projected onto a base of
integrable functions which are the Fourier functions. Let us denote Z ~ , p ~ and q ~ the Fourier transform of respectively
the height map and the X, Y components of the gradient field; µ and ν the reciprocal coordinates in the Fourier space.
Then the Fourier transform of the height map can be expressed as Equation (4) for each pixel.
* ~ *
~ − i.
µ .
~
p − i . ν . q
Z ( µ , ν ) =
(4)
2 2
µ + ν
Finally an inverse Fourier transform operation is computed in order to obtain the height map of the sample’s surface.
This algorithm is very efficient in term of both precision of the results and speed.
3.1 Illumination design
3. EXPERIMENTAL SETUP
As we measure the amount of polarized light reflected by the sample to be characterized, an unpolarized
illumination had to be designed. It is based on an integrating sphere which uses a white Lambertian coating. The light
source used is made of 2 green LEDs. These LEDs were chosen for several reasons: first of all, they provide high
luminous flux, typically 160 lm per LED. They have a narrow spectrum, which is an advantage as the polarization
elements used in the polarization camera presents some chromatic dependency of its polarization properties. Figure 3
shows pictures of the sphere.

Figure 3. Picture of the integrating sphere (a) Closed (b) Open with illumination
3.2 Polarization camera
The polarization camera used is composed of a fast polarization modulator developed by Bossa Nova Technologies
[16]. It allows switching polarization state between 4 linear polarization states (0°, 45°, 90° and 135°). The camera is
able to perform polarization imaging of a scene. With the 4 polarization images acquired in real time, it is easy to
calculate the linear Stokes parameters and the following polarization parameters - Degree of Linear Polarization (DOLP)
and the Angle of Polarization (AOP) – for each pixel of the image. The polarization camera is based on a standard CCD
camera. This CCD camera has good capacity, resolution, noise and linearity. Figure 4 shows the polarization camera
mounted vertically to image the sample that is mounted inside the integrating sphere.
Figure 4. (a) Bossa Nova Technologies linear Stokes camera SALSA TM (b) DOLP grayscale image of a 1’’ aspheric lens (c) AOP as a
color HUE
The DOLP is increasing with the radius, as the slope of the optic is; the AOP is varying according to the azimuth angle.
3.3 Sample mount
The measurement is based on the analysis of the light reflected by the first surface of the sample. To be able to
measure only the light reflected by the first surface, it is necessary to remove any other reflected and scattered light
coming from the back surface of the sample or from the holder. The sample needs to be both opaque and non scattering.
This can be achieved by index matching the back surface of lens to the support. In this case, the support must have the
same refractive index as the lens. The best to this end is to use an absorptive glass optical density and index matching
liquid (figure 5).

Figure 5: (a) the specular reflection at an air-glass interface is characterized by a 4% reflection coefficient. (b) Using an optical density
as a support avoids both transmitted and scattered light. (c) Index matching the optical density and the lens eliminates the surface
reflections
We used an index matching liquid of refractive index n=1.52. The refractive index of BK7 (most commonly used glass)
is exactly this value around 520nm (green). To test the efficiency of the set up, we imaged the lens on the absorptive
neutral density without index matching liquid, with water (refractive index 1.33) as index matching material and finally
with the index matching material (figure 6). Without index matching material, a strong back reflection is visible. It is
even brighter than the front reflection because it comes from the interface between the lens and air and from the interface
between the absorptive neutral density and air. Interferences fringes are visible on the back surface reflection (equal
thickness interference fringes). The reflections at the back surface are about to 8% of the total light which is twice as
much as the front surface reflection. With water as index matching material, the back surface reflections are highly
attenuated but are still not negligible in front on the front surface reflection. The total reflection coefficient is about
0.9%. With the index matching liquid, the back reflection totally disappears; only the front surface reflection is visible.
Figure 6: (a) lens on an absorptive optical density. The back surface reflection is stronger than front surface reflection. (b) Using water
as index matching material dramatically reduces back surface reflection. (c) Using a specific index matching liquid totally eliminates
the back surface reflection
3.4 Full Setup
Figure 7 shows a picture of the complete system, showing the polarization camera, the integrating sphere and the
sample.

Polarization camera
Integrating sphere
Sample
Illumination LEDs
Figure 7. Full Setup
4. TESTS ON COMMERCIAL SAMPLES - ANALYSIS
In order to evaluate the performances and the limitations of the developed system we realized many tests on several
spherical and aspheric commercial samples. Measurements were performed on spherical lenses with Radius of Curvature
ranging from 15mm to 110mm. Profiles can be deduced from the 3D shape and compared to theoretical values. The
repeatability of the measurement can be evaluated by subtracting two measurements. Figure 8 shows a screenshot of a
sample’s surface 3D reconstruction (a), the selection of a section profile to be plotted (b) and an extracted profile (c).
Figure 8. (a)Reconstructed surface, (b) section (c) extracted profile
4.1 Tests on commercial samples
Profile measurements
Profile measurements were performed on commercial highly aspheric lenses. These profiles were compared to
very accurate measured profiles (performed with 250nm resolution mechanical profilometer) given by the constructor in
order to evaluate the performances of the system. Figure 9 shows a profile measured on a commercial aspheric lens
sample with a high departure from sphere parameter (conic constant: k = -2.27, Departure From Sphere=180µm). The
red curve shows the profile acquired by the constructor’s contact profilometer (1nm resolution) and the black curve
shows the one acquired by our system.

Theoretical and BNT Profiles
0.5
0
-15 -10 -5 0 5 10 15
-0.5
-1
Height (mm)
-1.5
-2
-2.5
-3
Theoretical Profile
-3.5
BNT profile
-4
X section (mm)
Figure 9: Measured and theoretical profiles – aspheric sample
One can notice the strong artifact at the top of the surface. It is due to the reflection of the diaphragm on the sample
surface which modifies the DOLP value. The DOLP is already very low on the central area of the sample because the
incident angle of the light beam is very close to the normal direction. So the combination of those two parameters leads
to that strong artifact. Figure 10 shows the plot of the difference between the reconstructed and the theoretical profile. If
we forget the strong artifact at the center due to the reflection of the pupil lens on the sample we can notice that the error
is a slowly increasing in the field and has a rotational symmetry component. The maximum amplitude is around 25µm at
the edge of the sample. There is a strong field dependency. However, we think that dependency is an artifact and does
not show the real limitation of the reconstruction method. The real limitation is more likely to be the high frequency
noise around the slow variation. The amplitude of that noise is about 100nm which correspond to the order of magnitude
given by our previous simulations. In the following section we will analyze the possible sources of these artifacts.
Figure 10. Difference between the theoretical and the reconstructed profile

Repeatability
Several measurements in a row were performed on the same sample (removing the sample from the holder) in
order to evaluate the repeatability of the results. Reconstructed surfaces of two successive acquisitions are subtracted.
Residual noise and artifacts can be estimated. Figure 11 shows the 3D residual noise of two measurements on a 20.6mm
ROC sample (sample removed between the 2 measurements). The PV value of the residual surface is 1.9µm. This is
extremely encouraging as the measurements are done without any specific optical alignment like the ones required for
interferometric methods.
Figure 11. Measurement repeatability
5. CURRENT LIMITATIONS – FURTHER STEPS
These preliminary tests were very useful to estimate the limitations of the system. The main limitations are the
artifact due to the central reflection and the slow deviation observed which is increasing with the field of view. The
potential sources of errors investigated are listed below:
Artifacts in the 3D reconstruction algorithm itself
The measurement of DOLP and AOP parameters
5.1 Artifacts introduced by the algorithm
The point is to evaluate the error introduced by the reconstruction algorithm itself. We generated perfect DOLP and
AOP map and used the reconstruction algorithm. Then we can compare the reconstructed spheres with a theoretical
sphere and see the defects introduced by the algorithm. Then we have computed the difference between the reconstructed
shape and the theoretical one. Figure 12 shows the 3D plot and an extracted profile of that difference

Difference and fitting
0.00E+00
-1.50E+01 -1.00E+01 -5.00E+00 0.00E+00 5.00E+00 1.00E+01 1.50E+01
Amplitude of the difference (mm)
-5.00E-04
-1.00E-03
-1.50E-03
-2.00E-03
-2.50E-03
-3.00E-03
X section (mm)
Difference Perfect sphere -
Reconstructed shere
Fitting with a spherical profile
Figure 12: 3D plot (a) and profile (b) of the deviation between perfect sphere and reconstructed sphere
The PV amplitude of the deviation is around 2.7µm for a simulated 40 mm Radius Of Curvature. Moreover the profile
shape of the deviation appears to be spherical and using a numerical solver we can estimate the best sphere parameters
which fit the deviation profile (especially the Radius Of Curvature of the deviation profile). Figure 13 shows that the
ROC of the deviation profile is proportional to the ROC of the sphere to be reconstructed.
ROC of the deviation profile vs. ROC of
theoretical sphere
35000
30000
ROC defect (mm)
25000
20000
15000
10000
5000
0
0 10 20 30 40 50 60
ROC perfect sphere (mm)
Figure 13: ROC of the profile deviation vs. ROC of the sphere
These tests have shown that the algorithm itself introduces artifact in the 3D reconstructed shape. Those artifacts strongly
depend on the geometric parameters of the system. The PV amplitude of the defect ranges from 2 to 4µm depending on
the Radius Of Curvature of the sphere and on geometrical parameters. The reconstruction process assumes that the
distance between the entrance pupil and the sample as well as the magnification of the optical conjugation is uniform. In
reality, as shown in figure 14, these parameters vary in the field. This is particularly true with high sag optics such as the
tested aspherics. That explains why the ROC of the deviation profile is increasing with the ROC of the sphere to be
reconstructed. The more the ROC of the sphere is, the less the sag is, and the less the amplitude of the deviation is. The
consequences are a non uniform integration of the gradient field and could lead to such an artifact. The same kind of
error has been observed in the measurement of the DOLP parameter; it will be detailed in the following section.

∆d
Sample
Lens
Detector
Distance
Distance + ∆d
Figure 14 Non uniformity of the distance Sample – Entrance pupil
5.2 Measurement of DOLP and AOP parameters
Measured DOLP profiles were compared to theoretical DOLP. As shown on Figure 15 the measured DOLP are
always smaller than expected values while the field is increasing. The DOLP error is up to 0.11 at the edge (11% DOLP).
It has a strong impact on the final result as it is related to the local slope of the surface. That error is due to an important
amount of flare light in the setup.
Figure 15: (a) Measured and Theoretical DOLP profiles, 1’’ spherical commercial sample, ROC=20.6mm; (b) Deviation
If we call I 0 the intensity of the flare light, I S and I P respectively the intensity of s-polarized and p-polarized light,
DOLP mes and DOLP th respectively the measured and the theoretical DOLP, we can see the effect on the DOLP
deviation:
∆
∆
DOLP
DOLP
=
=
DOLP
I
S
measured
I
0
+ I + 2 I
P
− DOLP
0
× DOLP
theoretica l
2 (4)
th
= K × DOLP
theoretica l

That leads to a DOLP error proportional to the DOLP expected value. That assumption was verified experimentally. The
K constant depends only on the amount of flare light which is related to the experimental setup. Therefore it is possible
to measure it and then to correct systematically the DOLP error. That compensation was implemented and gave excellent
results. Figure 16 shows corrected DOLP and the new deviation. The error was put down 0.4% pic-to-valley.
Figure 16 : Calibrated an theoretical DOLP, 1’’ spherical commercial sample, ROC=20.6mm; (b) Deviation
Similar results have been observed on commercial spherical and aspheric optics with Radius Of Curvature ranging
between 15mm to 105mm. The DOLP correction performed here does not take into account any field dependency. The
next step should be to perform a complete calibration of DOLP and AOP values using an angle controlled reflecting
plane. Then we could build a calibration matrix for each angle. The real noise appears as a high frequency modulation of
the symmetric remaining artifact. Its amplitude is around 100nm.
6. CONCLUSION
The principle of a polarization imaging system for 3D reconstruction of optical elements was successfully
demonstrated. A complete setup based on an unpolarized illumination and Bossa Nova Technologies’ linear Stokes
polarization camera was designed. A dedicated reconstruction algorithm based on the off-the-shelf Frankot-Chellapa
algorithm was developed and tested on several commercial samples. The described setup performances were evaluated
by comparing measured section profiles and theoretical profiles given by constructors. The samples tested with the
method were one and two inches diameters, spherical and aspheric, with Radius of Curvature ranging from 15mm to
110mm and Departure From Sphere ranging from 0 to 800µm. The best accuracy obtained was 25µm with this setup.
This is very encouraging as the current setup was based on relatively cheap components and no specific calibrations or
signal/image processing was performed.
This work has been sponsored by a SBIR phase I grant from the National Science Foundation.
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